Product Hardy spaces associated to operators with heat kernel bounds on spaces of homogeneous type
Recent Developments in Harmonic Analysis May 15, 2017  May 19, 2017
Location: SLMath: Eisenbud Auditorium
harmonic analysis
product Hardy spaces
heat kernel bounds
CalderonZygmund theory
multiplier operators
35J58  Boundary value problems for higherorder elliptic systems
46E10  Topological linear spaces of continuous, differentiable or analytic functions
30B70  Continued fractions; complexanalytic aspects [See also 11A55, 40A15]
Ward
Much effort has been devoted to generalizing the Calder\'onZygmund theory from Euclidean spaces to metric measure spaces, or spaces of homogeneous type. Here the underlying space $\mathbb{R}^n$ with Euclidean metric and Lebesgue measure is replaced by a set $X$ with a general metric or quasimetric and a doubling measure. Further, one can replace the Laplacian operator that underpins the Calder\'onZygmund theory by more general operators~$L$ satisfying heat kernel estimates. I will present recent joint work with P.~Chen, X.T.~Duong, J.~Li and L.X.~Yan along these lines. We develop the theory of product Hardy spaces $H^p_{L_1,L_2}(X_1 \times X_2)$, for $1 \leq p < \infty$, defined on products of spaces of homogeneous type, and associated to operators $L_1$, $L_2$ satisfying DaviesGaffney estimates. This theory includes definitions of Hardy spaces via appropriate square functions, an atomic Hardy space, a Calder\'onZygmund decomposition, interpolation theorems, and the boundedness of a class of Marcinkiewicztype spectral multiplier operators.
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